MECH 314 Notes 7, Spring 2014

MECH 314 Introduction to Fluid Mechanics April 26, 2009 American University of Beirut, Spring 2009 Handout # 5 Incompressible Flow Criterion A flow is incompressible if ����� 1 ρ dρ dt ����� << ����� ∂u ∂x ����� + ����� ∂v ∂y ����� + ����� ∂w ∂z ����� or ����� dρ ρ ����� << ������ ∂u ∂x ����� + ����� ∂v ∂y ����� + ����� ∂w ∂z ����� � dt The term on the right hand side scales as one, since |∂u/∂x| ∼ U/L and δt ∼ L/U, where L and U are respectively characteristic length and velocity. So a flow is approximated as incompressible if density change δρ over an infinitesimal process t to t + δt is very small compared to the density ρ. Under the condition ����� δρ ρ ����� << 1 ⇒ ∇ · u ≃ 0 For a simple single-phase fluid, the thermodynamic state is defined by two independent variables. Choosing pressure and specific entropy as these independent variables, then ρ = ρ(p, s) ⇒ dρ = �∂ρ ∂p � s dp + �∂ρ ∂s � p ds The term � ∂ρ ∂p � s, denoting the change in density cause by change in pressure during an infinitesimal isentropic process, is the inverse of the speed of sound (a) in the fluid squared �∂ρ ∂p � s ≡ 1 a2 ⇒ δρ = δp a2 So for an isentropic infinitesimal process, the following condition for incompressibility must hold ����� δp ρ ����� << a2 (1) 1 For the flow to be isentropic, it must necessarily be inviscid (why?), so that du dt = −1 ρ∇p + g so that U (L/U) ∼ δp/L ρ ⇒ ����� δp ρ ����� ∼ U 2 Combining with eq. (1), the condition for incompressibility for an isentropic infinitesimal process reduces to U 2 a2 << 1 incompressible flow condition Recognizing the ratio of flow speed to speed of sound in the fluid as Mach number Ma, then Ma < 0.2 incompressible flow condition Challenge Show that the speed of sound in an ideal gas is a = √γRT, where γ = cp/cv. 2